# Mathematical and Physical Geometry

This page is a sub-page of our page on Physics and its Models.

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Hans Reichenbach (1958 (1927)) The Philosophy of Space and Time, Dover, ISBN 0-486-60443-8.

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/////// Quoting Reichenbach: Philosophy of Space and Time

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/////// Quoting Reichenbach on coordinative definitions (1957 (1927), p. 15):

Defining usually means reducing a concept to other concepts. In physics, as in all other fields of inquiry, wide use is made of this procedure. There is a second kind of definition, however, which is also employed and which derives from the fact that physics, in contradistinction to mathematics, deals with real objects.

Physical knowledge is characterized by the fact that concepts are not only defined by other concepts, but are also coordinated to real objects. This simply states that this concept is coordinated to this particular thing. In general this coordination is not arbitrary. Since the concepts are interconnected by testable relations, the coordination may be verified as true or false, if the requirement of uniqueness is added, i.e., the rule that the same concept must always denote the same object.

The method of physics consists in establishing the uniqueness of this coordination, as Schlick has clearly shown. But certain preliminary coordinations must be determined before the method of coordination can be carried through any further; these first coordinations are therefore definitions which we shall call coordinative definitions. They are arbitrary, like all definitions; on their choice depends the conceptual system which develops with the progress of science.

Wherever metrical relations are to be established, the use of coordinative definitions is conspicuous. If a distance is to measured, the unit of length has to be determined beforehand by definition. This definition is a coordinative definition. Here the duality of conceptual definition and coordinative definition can easily be seen. We can define only by means of other concepts what we mean by a unit; for instance: “A unit is a distance which, when transported along another distance, supplies the measure of this distance.” But this statement does not say anything about the size of of the unit, which can only be established by reference to a physically given length such as the standard meter in Paris. […]

The characteristic feature of this method is the coordination of a concept to a physical object. These considerations explain the term “coordinative definition.” If the definition is used for measurements, as in the case of the unit of length, it is a metrical coordinative definition.

The philosophical significance of the theory of relativity consists in the fact that it has demonstrated the necessity for metrical coordinative definitions in several places where empirical relations had previously been assumed. It is not always as obvious as in the case of the unit of length that a coordinative definition is required before any measurements can be made, and pseudo-problems arise if we look for truth where definitions are needed. The word “relativity” is intended to express the fact that the results of the measurements depend upon the choice of the coordinative definitions. It will be shown presently how this idea affects the solution of the problem of geometry.

After this solution of the problem of the unit of length, the next step leads to the comparison of two units of length at different locations. If the measuring rod is laid down, its length is compared only to that part of a body, say a wall, which it covers at the moment. If two separate parts of the wall are to be compared, the measuring rod will have o be transported. It is assumed that the measuring rod does not change during the transport. It is fundamentally impossible, however, to detect such a change if it is produced by universal forces. […]

The problem does not concern a matter of cognition but of definition. There is no way of knowing whether a measuring rod retains its length when it is transported to another place; a statement of this kind can only be introduced by a definition. For this purpose a coordinative definition is to be used, because two physical objects distant from each other are defined as equal in length. It is not the concept equality of length which is to be defined, but a real object corresponding to it is to be pointed out. A physical structure is coordinated to the concept equality of length, just as the standard meter is coordinated to the concept unit of length.

This analysis reveals how definitions and empirical statements are interconnected. As explained above, it is an observational fact, formulated in an empirical statement, that two measuring rods which are shown to be equal in length by local comparison made at a certain space point will be found equal in length by local comparison at every other space point, whether they have been transported along the same or different paths. When we add to this empirical fact the definition that the rods shall be called equal in length when they are at different places, we do not make an inference from the observed fact; the addition constitutes an independent convention.

There is, however, a certain relation between the two. The physical fact makes the convention unique, i.e., independent of the path of transportation. The statement about the uniqueness of the convention is therefore empirically verifiable and not a matter of choice. One can say that the factual relations holding for a local comparison of rods, though they do not require the definition of congruence in terms of transported rods, make this definition admissible. Definitions that are not unique are inadmissible in a scientific system.

This consideration can only mean that the factual relations may be used for the simple definition of congruence where any rigid measuring rod establishes the congruence. If the factual relations did not hold, a special definition of the unit of length would have to be given for every space point. Not only at Paris, but also at every other place a rod having the length of a “meter” would have to be displayed, and all these arbitrarily chosen rods would be called equal in length by definition. The requirement of uniformity would be satisfied by carrying around a measuring rod selected at random for the purpose of making copies and displaying these as the unit. If two of these copies were transported and compared locally, they would be different in length, but this fact would not “falsify” the definition.

In such a world it would become very obvious that the concept of congruence is a definition; but we, in our simple world, are also permitted to choose a definition of congruence that does not correspond to the actual behaviour of rigid rods. Thus we could arrange measuring rods, which in the ordinary sense are called equal in length, and, laying them end to end call the second rod half as long as the first, the third one a third etc. Such a definition would complicate all measurements, but epistemologically it is equivalent to the ordinary definition, which calls the rods equal in length.

In this statement we make use of the fact that the definition of a unit at only one space point does not render general measurements possible. For the general case the definition of the unit has to be given in advance as a function of the place (and also of the time). It is again a matter of fact that our world admits of a simple definition of congruence because of the factual relations holding for the behaviour of rigid rods; but this fact does not deprive the simple definition of its definitional character.

The great significance of the realization that congruence is a matter of definition lies in the fact that by its help the epistemological problem of geometry is solved. The determination of the geometry of a certain structure depends on the definition of congruence. […] The geometrical form of a body is no absolute datum of experience, but depends on a preceding coordinative definition; depending on the definition, the same structure may be called a plane, or a sphere, or a curved surface. […]

We are now left with the problem which coordinative definition should be used for physical space? Since we need a geometry, a decision has to be made for a definition of congruence. Although we must do so, we should never forget that we deal with an arbitrary decision that is neither true nor false. Thus the geometry of physical space is not an immediate result of experience, but depends on the choice of the coordinative definition.

In this connection we shall look for the most adequate definition, i.e., the one which has the advantage of logical simplicity and requires the least possible change in the results of science. The sciences have implicitly employed such a coordinative definition all the time, though not always consciously; the results based upon this definition will be developed further in our analysis. It can be assumed that the definition hitherto employed possesses certain practical advantages justifying its use. In the discussion about the definition of congruence by means of rigid rods, the coordinative definition has already been indicated. The investigation is not complete, however, because an exact definition of the rigid body is still missing.

////// End of quote from Reichenbach

/////// Quoting Reichenbach on Rigid Bodies (Reichenbach, 1957 (1927), p. 19):

Experience tells us that physical objects assume different states. Solid bodies have an advantage over liquid ones because they change their shape and size only very little when affected by outside forces. They seem, therefore, to be useful for the definition of congruence. However, if the result of the previous considerations is kept in mind, this relative stability is no ground on which to base a preference for solid bodies.

As was explained, the form and size of an object depends on the coordinative definition of congruence; if the solid body is used for the coordinative definition, the statement that it does not change its shape must not be regarded as a cognitive statement. It can only be a definition: we define the shape of the solid body as unchangeable. But how can a sloid body be defined? In other words, if the physical state of being solid were defined differently, under what conditions would the solid body be called rigid? If the conservation of shape is not permissible as a criterion, what criteria may be used?

The problem becomes more complicated because we cannot solve it by merely pointing to certain real objects. Although the standard meter in Paris was cited previously as the prototype of such a definition, this account was a somewhat schematic abstraction. Actually no object is the perfect realization of the rigid body of physics; it must be remembered that such an object may be influenced by many physical forces. Only after several corrections have been made, for example, for the influence of temperature and elasticity, is the resulting length of the object regarded as adequate for the coordinative definition of the comparison of lengths.

The standard meter in Paris would not be accepted as the definition of the unit of length, if it were not protected from influences of temperature, etc., by being kept in a vault. If an earthquake should ever throw it out of this vault and deform its diameter, nobody would want to retain it as the prototype of the meter; everybody would agree that the standard meter would no longer be a meter. But what kind of definition is this, if the definition may some day be called false? Does the concept of coordinative definition become meaningless?

The answer is: it does not become meaningless, but, as we shall see, its application is logically very complicated. The restrictions that affect the arbitrariness of the coordinative definition have two sources. One restriction lies in the demand that the obtained metric retain certain older physical results, especially those of the “physics of daily life”. Nobody could object on logical grounds if the bent rod would be taken as the definition of the unit of length; but then we must accept the consequence that our house, our body, the whole world has become larger. Relative to the coordinative definition it has, indeed, become larger, but such an interpretation does not correspond to our habitual thinking. We prefer an interpretation of changes involving an individual thing on the one side and the rest of the world on the other side that confines the change to the small object.

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/////// End of quote from Reichenbach

/////// Quoting Reichenbach on The Relativity of Geometry (Reichenbach, (1957 (1927), pp.30-37)):

With regard to the problem of geometry we have come to realize that the question which geometry holds for physical space must be decided by measurements, i.e., empirically. Furthermore, this decision is dependent on the assumption of an arbitrary coordinative definition of the comparison of length. […]

Our conception permits us to start with the assumption that Euclidean geometry holds for physical space. Under certain conditions, however, we obtain the result that there exists a universal force $\, F \,$ that deforms all measuring instruments in the same way. However, we can invert the interpretation: we can set $\, F \,$ equal to zero by definition and correct in turn the theory of our measuring instruments. We are able to proceed in this manner because a transformation of all measurements from one geometry into another is possible and involves no logical difficulties. […]

It turns out that the non-Euclidean geometry obtained under our coordinative definition of the rigid body deviates quantitatively only very little from Euclidean geometry when small areas are considered. In this connection, “small areas” means “on the order of the size of the earth”; deviations from Euclidean geometry can be noticed only in astronomic dimensions. […]

The objection is connected with the a priori theory of space that goes back to Kant and today is represented in various forms. Not only Kantians and Neo-Kantians attempt to maintain the a priori character of geometry: the tendency is also pronounced in philosophical schools which in other respects are non-Kantian. […] In the course of the discussion of the theory of relativity, it has become evident that the philosophy of Kant has been subject to so many interpretations by his disciples that it can no longer serve as a sharply defined basis for present day epistemological analysis. […]

Mathematics proves that every geometry of the Riemannian kind can be mapped upon another one of the same kind. In the language of physics this means the following:

Theorem 1: Given a geometry $\, G \,$ to which the measuring instruments conform, we can imagine a universal force $\, F \,$ which affects the instruments in such a way that the actual geometry is an arbitrary geometry $\, G' \,$, while the observed deviation from $\, G \,$ is due to a universal deformation of the measuring instruments.

No epistemological objection can be made against the correctness of Theorem 1. Is the visual a priori compatible with it? Offhand we must say yes. Since the Euclidean geometry $\, G_0 \,$ belongs to the geometries of the Riemannian kind, it follows from Theorem 1 that it is always possible to carry through the visually preferred geometry for physical space. Thus we have proved that we can always satisfy the requirements of visualization.

But something more is proved by Theorem 1 which does not fit very well into the theory of the visual a priori. The theorem asserts that Euclidean geometry is not preferable on epistemological grounds. Theorem 1 shows all geometries to be equivalent; it formulates the principle of relativity of geometry. It follows that it is meaningless to speak about one geometry as the true geometry. We obtain a statement about physical reality only if in addition to the geometry $\, G \,$ of the space its universal field of force $\, F \,$ is specified. Only the combination $\, G + F \,$ is a testable statement.

We can now understand the significance of a decision for Euclidean geometry on the basis of the visual a priori. The decision means only the choice of a specific coordinative definition. In our definition of the rigid body we set $\, F = 0$; the statement about the resulting $\, G \,$ is then a univocal description of reality. This definition means that in $\, "G + F" \,$ the second term is zero. The visual a priori, however, sets $\, G = G_0$. But then the empirical component in the results of measurement is represented by the determination of $\, F$; only through the combination $\, G_0 + F$ are the properties of space exhaustively described. […]

Geometry is concerned solely with the simplicity of a definition, and therefore the problem of empirical significance does not arise. It is a mistake to say that Euclidean geometry is “more true” than Einstein’s geometry or vice versa, because it leads to simpler metrical relations. We said that Einstein’s geometry leads to simpler metrical relations because in it $\, F = 0$.

But we can no more say that Einstein’s geometry is “truer” than Euclidean geometry, than we can say that the meter is a “truer” unit of length than the yard. The simpler system is always preferable; the advantage of meters and centimeters over yards and feet is only a matter of economy and has no bearing upon reality. Properties of reality are discovered only by a combination of the results of measurement with the underlying coordinative definition.

The same affairs can therefore be described in different ways. In our example, it could just as well be said that in the neighborhood of a heavenly body a universal field of force exists which affects all measuring rods, while the geometry is Euclidean. Both combinations of statements are equally true, as can be seen from the fact that one can be transformed into the other. Similarly, it is just as true to say that the circumference of the earth is 40 million meters as to say that it is 40 thousand kilometers. The significance of this simplicity should not be exaggerated; this kind of simplicity, which we call descriptive simplicity, has nothing to do with truth.

Taken alone, the statement that a certain geometry holds for space is therefore meaningless. It acquires meaning only if we add the coordinative definition used in the comparison of widely separated lengths. The same rule holds for the geometrical shape of bodies. The sentence “the earth is a sphere” is an incomplete statement, and resembles the statement “This room is seven units long”. Both statements say something about objective states of affairs only if the assumed coordinative definitions are used. These considerations indicate what is meant by relativity of geometry.

This conception of the problem of geometry is essentially the result of the work of Riemann, Helmholtz and Poincaré and is known as conventionalism.

While Riemann prepared the way for an application of geometry to physical reality by his mathematical formulation of the concept of space, Helmholtz laid the philosophical foundations. In particular, he recognized the connection of the problem of geometry with that of rigid bodies and interpreted correctly the possibility of a visual representation of non-Euclidean spaces. (cf. p. 63). It is his merit, furthermore, to have clearly stated that Kant’s theory of space is untenable in view of recent mathematical developments. Helmholtz’ epistemological lectures must therefore be regarded as the source of modern philosophical knowledge of space.

It is Einstein’s achievement to have applied the theory of relativity of geometry to physics. The surprising result was the fact that the world is non-Euclidean, as the theorists of relativity are wont to say; in our language this means: if $\, F = 0$, the geometry $\, G \,$ becomes non-Euclidean. This outcome had not been anticipated, and Helmholtz and Poincaré still believed that the geometry obtained could not be proved to be different from Euclidean geometry. Only Einstein’s theory of gravitation predicted the non-Euclidean result which was confirmed by astronomical observation. The deviations from Euclidean geometry, however, are very small and not observable in everyday life.

Unfortunately, the philosophical discussion of conventionalism, misled by its ill-fitting name, did not always present the epistemological aspect of the problem with sufficient clarity. From conventionalism the consequence was derived that it is impossible to make an objective statement about the geometry of physical space, and that we are dealing with subjective arbitratiness only; the concept of geometry of real space was called meaningless. This is a misunderstanding. Although the statement about the geometry is based upon certain arbitrary definitions, the statement itself does not become arbitrary: once the definitions have been formulated, it is determined through objective reality alone which is the actual geometry.

Let us use our previous example: although we can define the scale of temperature arbitrarily, the indication of the temperature of a physical object does not become a subjective matter. By selecting a certain scale we can stipulate a certain arbitrary number of degrees of heat for the respective body, but this indication has an objective meaning as soon as the coordinative definition of the scale is added. On the contrary, it is the significance of coordinative definitions to lend an objective meaning to physical measurements.

As long as it was not noticed at what points of the metrical system arbitrary definitions occur, all measuring results were undetermined; only by discovering the points of arbitrariness, by identifying them as such, and by classifying them as definitions can we obtain objective measuring results in physics. The objective character of the physical statement is thus shifted to a statement about relations. A statement about the boiling point of water is no longer regarded as an absolute statement, but as a statement about a relation between the boiling water and the length of the column of mercury.

There exists a similar objective statement about the geometry of real space: it is a statement about a relation between the universe and rigid rods. The geometry chosen to characterize this relation is only a mode of speech; however, our awareness of the relativity of geometry enables us to formulate the objective character of a statement about the geometry of the physical world as a statement about relations. In this sense we are permitted to speak of physical geometry. The description of nature is not stripped of arbitrariness by naïve absolutism but only by recognition and formulation of the points of arbitrariness. The only path to objective knowledge leads through conscious awareness of the role that subjectivity plays in our methods of research.

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